Quantum simulation of Motzkin spin chain with Rydberg atoms

Abstract

Motzkin spin chain is a well-known mathematical model with connections to symmetry-protected topological phases, such as the Haldane phase, as well as to concepts in the AdS/CFT correspondence. They exhibit highly entangled ground states that violate the area law and are exceptionally difficult to simulate with conventional numerical methods. Numerical simulations of the Motzkin ground state become further challenging at large system sizes due to their high-dimensional spin structure, rendering it a natural test bed for quantum simulation with ultra-cold systems. Here, we propose a Rydberg-atom based quantum simulation scheme that effectively realizes Motzkin spins using an experimentally accessible set of parameters. We show that the resulting effective Motzkin ground state reproduces the characteristic entanglement scaling and the block-structure properties of the reduced density matrix associated with the ideal Motzkin state. Our results establish a pathway toward a concrete experimental realization of Motzkin spins beyond purely mathematical constructions, opening avenues for exploring other similar exotic non-area-law entangled phases in programmable Rydberg simulators.

Figures (10)

• Figure 1: (a) An example of a Motzkin path with respective encoding of each path with the three levels of spin-1 particle, denoted by $\{\ket{\uparrow},\ket{0},\ket{\downarrow}\}$. (b) Encoding the three levels of spin-1 particle into Rydberg atoms with principal quantum numbers $\nu$ and $\nu'$, and angular quantum numbers $s$$(l=0)$ and $p$$(l=1)$. Red-dashed lines and green-dotted arrows indicate spin-exchange dipole-dipole interactions and van der Waals based energy shifts, respectively, while the blue arrows indicate Förster resonance interactions.
• Figure 2: (a) Motzkin and (b) Inverse Motzkin states, for $N=3$, shown in both spin and path representations. Motzkin paths remain non-negative at all steps, whereas inverse-Motzkin paths violate this constraint and form the vertical mirror counterparts of Motzkin paths.
• Figure 3: Adiabatic-control protocol: (1) state initialization, (2) preparation of the Rydberg ground state, (3) application of a slowly-varying locally controlled microwave coupling denoted by $\Omega_{1,3}^{\uparrow,\downarrow},\delta_{1,3}^{\uparrow,\downarrow}$ and (4) emergence of the effective Motzkin state.
• Figure 4: (a) Population dynamics of the $N=2$ Rydberg system, showing the suppression of undesired states and the emergence of the effective Motzkin ground state. (b) Fidelity $\mathcal{F}$ of the effective ground state with respect to the Motzkin ground state for system sizes $N = 2$ to $8$.
• Figure 5: System size scaling of entanglement entropy (a) von Neuman entropy ($S_1$) and (b) Rényi entropy of order 2 ($S_2$), of effective Motzkin ground state (blue-solid), ideal Motzkin ground state (red-dashed) and Rydberg ground state (black-dot-dashed), for subsystem sizes $N_A=N/2$. The shade of the blue markers encodes the fidelity of the effective Motzkin state, with darker shades corresponding to higher fidelity.